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A straightforward way to improve the payload capacity and flight duration of a multirotor aerial vehicle is by combining it with a balloon filled with a lifting gas, such as helium or hydrogen. The balloon provides a net aerostatic lift that is oriented contrary to the vehicle weight, thus reducing the equivalent load supported by the multirotor airframe. The present work is concerned with the dynamic modeling as well as the design of position and attitude control laws for a balloon-multirotor vehicle consisting of an oblate spheroid helium balloon coupled with a hexacopter structure. A six-degrees-of-freedom nonlinear dynamic model is derived for the balloon-hexacopter using the Newton-Euler approach and considering, among other common efforts, a restoration torque due to the displacement of the balloon's center of buoyancy above the vehicle's center of mass. In order to capture the contact flexibility between the balloon and the airframe, the center of buoyancy is supposed to oscillate with a second-order dynamics with respect to the airframe. Under the assumption of time-scale separation between the translational and rotational dynamics, the attitude and position control laws are designed separately from each other. Both the attitude and position control laws are proportional-derivative actions plus nonlinear feedforward terms for feedback linearization combined with control input saturation within appropriate parallelepipedal sets. These constraint sets are carefully chosen in order to satisfy torque, force and inclination design bounds. A parametric probabilistic approach where temperature and pressure are modeled as uniform-distributed random variables, and propagation of uncertainties is computed via Monte Carlo method, is employed to assess the overall system performance. Simulation results are presented to demonstrate the performance and investigate some properties of the proposed balloon-hexacopter control system.